Synchronization and Balancing around Simple Closed Polar Curves with Bounded Trajectories and Control Saturation

TL;DR

This paper develops distributed control laws for multi-agent unicycle systems to synchronize and balance around simple closed polar curves, ensuring bounded trajectories, adherence to control saturation, and explicit boundary characterization.

Contribution

It introduces a novel control approach combining barrier Lyapunov functions with curve-phase potentials for bounded, saturated control around closed curves.

Findings

Agents asymptotically stabilize to the desired curve

Trajectories remain within a bounded set

Explicit boundary of the trajectory-constraining set is characterized

Abstract

The problem of synchronization and balancing around simple closed polar curves is addressed for unicycle-type multi-agent systems. Leveraging the concept of barrier Lyapunov function in conjunction with bounded Lyapunov-like curve-phase potential functions, we propose distributed feedback control laws and show that the agents asymptotically stabilize to the desired closed curve, their trajectories remain bounded within a compact set, and their turn-rates adhere to the saturation limits. We also characterize the explicit nature of the boundary of this trajectory-constraining set based on the magnitude of the safe distance of the exterior boundary from the desired curve. We further establish a connection between the perimeters and areas of the trajectory-constraining set with that of the desired curve. We obtain bounds on different quantities of interest in the post-design analysis and provide simulation results to illustrate the theoretical findings.